I want to introduce the slope of this kind of beam as the sum of the angle due to bending and angle due to the shear. I know that the typical stiffness matrix for this case dimensions 2x2.
Yes you can. To write the motion equations of a Timoshenko beam you need three degrees of freedom, two displacements (x,y) and one rotation (around z), then the stifness matrix is a 3x3 matrix.
Ok, but I need that mentioned degrees of freedom are:
- transverse displacement,
- angle (from bending),
- angle (from shear).
I think I found the answer to the question in the following paper:
D. L. THOMAS, J. M.WILSONANDR. R. WILSON, "TIMOSHENKO BEAM FINITE ELEMENTS", Journal of Sound and Vibration (1973) 31(3), 315-330
You do not need two angles, just one angle, it serves for measuring shear, bending is proportional to the derivative of the displacement. Note that geometrically exact theories in 3D require 6 parameters, three displacements and three rotations. Check this paper, it shows and easy 3d formulation with shear, you can obtain a 2D formulation eliminating one coordinate.
https://www.researchgate.net/publication/241105119_Free_vibration_and_dynamic_stability_of_rotating_thin-walled_composite_beams?ev=prf_pub Check the works of
You could also refert to Reissner works to clarify this aspects
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Article Free vibration and dynamic stability of rotating thin-walled...
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